<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29_1.html">
\begin{equation*}
\frac{\partial M}{\partial y}=3 x+2 y,\quad \frac{\partial N}{\partial x}=2 x+y,
\end{equation*}
</div>
<p class="continuation">Since <span class="process-math">\(\frac{\partial M}{\partial y}\neq \frac{\partial N}{\partial x}\text{,}\)</span> the given equation is not exact. Let’s see whether there exists a function <span class="process-math">\(\Psi(x, y)\)</span> which satisfies</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29_1.html">
\begin{equation}
\frac{\partial \Psi}{\partial x}=M(x, y)=3 x y+y^2,\quad \frac{\partial \Psi}{\partial y}=N(x, y)=x^2+x y.\tag{2.6.6}
\end{equation}
</div>
<p class="continuation">Integrating the first equation above, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29_1.html">
\begin{equation*}
\Psi(x, y)=\frac{3}{2} x^2 y+x y^2+h(y).
\end{equation*}
</div>
<p class="continuation">Then from <span class="process-math">\((\ref{eq2_29_1})_2\text{,}\)</span> one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29_1.html">
\begin{equation*}
\frac{\partial \Psi}{\partial y}=\frac{3}{2} x^2+2 x y+\frac{\textrm{d} h(y)}{\textrm{d} y}=N(x, y)=x^2+x y,
\end{equation*}
</div>
<p class="continuation">which implies</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_29_1.html">
\begin{equation*}
\frac{\textrm{d} h(y)}{\textrm{d} y}=-\frac{1}{2} x^2-xy.
\end{equation*}
</div>
<p class="continuation">Since the right hand side depends on <span class="process-math">\(x\)</span> as well, contradiction appears. So there is no <span class="process-math">\(\Psi(x, y)\)</span> satisfies (<a href="" class="xref" data-knowl="./knowl/eq2_29_1.html" title="Equation 2.6.6">(2.6.6)</a>).</p>
<span class="incontext"><a href="sec2_6.html#p-50" class="internal">in-context</a></span>
</body>
</html>
